Problem 25 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int t\left(t^{2}-1\right)^{5} d t $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{(t^2-1)^6}{12} + C \).

Step 1: Choose a Substitution

To solve the integral \( \int t(t^2-1)^5 \, dt \), we choose the substitution \( u = t^2 - 1 \). This choice simplifies the expression under the integral.

Step 2: Differentiate to Find du

Differentiate \( u = t^2 - 1 \) with respect to \( t \) to find \( du \). This gives \( du = 2t \, dt \). Solving for \( t \, dt \) gives \( t \, dt = \frac{1}{2} du \).

Step 3: Substitute and Simplify the Integral

Substitute \( u = t^2 - 1 \) and \( t \, dt = \frac{1}{2} du \) into the original integral. The integral becomes \( \int t(t^2-1)^5 \, dt = \int \frac{1}{2} u^5 \, du \). Simplifying, we have \( \frac{1}{2} \int u^5 \, du \).

Step 4: Integrate with Respect to u

Integrate \( \int u^5 \, du \), which is \( \frac{u^6}{6} + C \). Therefore, \( \frac{1}{2} \int u^5 \, du = \frac{1}{2} \times \frac{u^6}{6} = \frac{u^6}{12} + C \).

Step 5: Substitute Back to t

Replace \( u \) with \( t^2 - 1 \) to get the solution in terms of \( t \). This gives \( \frac{(t^2-1)^6}{12} + C \).

Step 6: Differentiate to Check the Solution

Differentiate \( \frac{(t^2-1)^6}{12} + C \) with respect to \( t \) to verify the integral. Using the chain rule, the derivative is \( \frac{1}{12} \times 6(t^2-1)^5 \times 2t = t(t^2-1)^5 \), which matches the original integrand.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus used to determine the rate at which a function is changing at any given point. It involves finding the derivative of a function. The derivative provides information about the slope of the function and helps us understand how changes in one variable affect changes in another. In our problem, we differentiate to check that the integration process was done correctly by reverting back to the original function.
For example, in the solution to our exercise, after finding the integral, we differentiate the result to ensure that it matches the original integrand. This involves applying basic differentiation rules to the expression, in this case:

• Recognizing that the function involves a power of a binomial expression.
• Applying differentiation techniques to simplify the process.

To confirm, the derivative of the solution \[ \frac{(t^2-1)^6}{12} + C \] is taken with respect to \(t\), revealing steps where various rules like the chain rule are applied to ensure accuracy.

Chain Rule

The chain rule is an essential tool in differentiation, especially when dealing with compositions of functions. It allows us to take the derivative of a composite function. To put it simply, if you have a function that is inside another function, the chain rule helps you differentiate both effectively.
In the context of our step-by-step solution, the chain rule is crucial for both the integration process and confirming the correctness of our solution. While checking the solution, the chain rule helps us differentiate the function \[ (t^2 - 1)^6 \]. This involves:

• Differentiating the outer function: Treat \( (t^2 - 1)^6 \) as a single variable raised to a power.
• Differentiating the inner function: Take the derivative of \( t^2 - 1 \).
• Combining the two results: Multiply the derivative of the outer function by the derivative of the inner function. This results in multiplying by the derivative \( 6(t^2 - 1)^5 \) with \(2t\).

This process reduces complex differentiation tasks into manageable steps by breaking them into simpler functions, illustrating the utility of the chain rule.

Indefinite Integral

An indefinite integral, also known as an antiderivative, is the reverse process of differentiation. It looks to find a function whose derivative matches a given function. Indefinite integrals are critical because they help solve a wide array of problems involving accumulation, areas under curves, among others.
In the exercise provided, we are tasked with solving \[ \int t(t^2-1)^5 \, dt \],representing the indefinite integral with respect to \(t\). This involves:

• Choosing a substitution: simplifies the integration process when a function is composed of other functions.
• Transforming the integration variable: Utilizing the substitution to transform the integral into a simpler form.
• Integrating with respect to the new variable: Applying integration rules to simplify further.

After integrating, the solution is an expression \[ \frac{u^6}{12} + C \],where you substitute back the original variable. This emphasizes finding a general form of the antiderivative inclusive of the constant \(C\), which accounts for any constant function that could shift the antiderivative vertically.